An axiomatic framework for three-way clustering

Three-way clustering provides a variety of models with richer structural features than traditional two-way clustering. However, the structural properties of three-way clustering have not been systematically studied. In this paper, we propose an axiomatic framework to study three-way clustering based on the structural properties of three-way clusters. We categorize three-way clustering models into 16 types organized in five levels according to six axioms. We propose three strategies for three-way clustering approaches: the 2to3WC Strategy, the I3WC Strategy, and the E3WC Strategy. These strategies comprehensively cover existing three-way clustering models. We examine each of the existing methods and incorporate almost all of them into these three strategies. The framework not only summarizes the structural properties and methods of existing studies but also provides inspiration for future research. There are six types of three-way clustering models that have not yet been proposed and require further study. Strategies that have not been applied to each type also suggest possible future research directions.